In Alex's Adventures in Numberland, John Horton Conway's concept ofpowertrains is mentioned on p.261.

For any number written abcdefg..., its 'powertrain' is (a^b).(c^d).(e^f).g...,with 0^0 = 1. So 2015 -> (2^0).(1^5) = 1.1 = 1.

Repeatedly reapplying a powertrain reduces almost every number to a single digit.Those which don't are called 'indestructible digits'.

Two numbers are cited as being indestructible, 2592, found by Conway, and 24547284284866560000000000, found by Neil Sloane. These are the only twoknown to exist.

2592 -> (2^5).(9^2) = 32.81 = 2592.

So far so good. But what about 2534?

2534 -> (2^5).(3^4) = 32.81 = 2592 -> 2592.

Doesn't that also make 2534 indestructible according to the definition,or have I misunderstood something?

Does the word 'reduce' mean that only powertrains with successively smalleroutputs are permissible? From the text, the implication is that this isn'tthe case. Numbers ending in 99 with an even number of digits all outputmuch higher numbers before collapsing to a single digit.

I have read same book and I was also wondering at same problem. It was some years ago, but now, when I am older I used my computer and made program to calculate this phenomenon. John Horton Conway and his friend mentioned only about these two numbers 2592 and 24547284284866560000000000 because they are undestroyable at first time of powertrain use (sorry for my English, it's not my native language). As I said I made a program what can find all no-powertrain numbers as far as long it continue calculating. There are some of them:

I think there's infinity numbers what are invulnerable for powertrain with more than one use. All these numbers goes to 2592, and maybe, if I'll find higher numbers than 24547284284866560000000000 probably they also reduces to that number, not only 2592. But these two numbers absolutely stands out.